Building proofs: a practical guide
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2015
|
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | X, 164 S. Ill., graph. Darst. |
| ISBN: | 9789814641296 9789814641302 |
Internformat
MARC
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| 020 | |a 9789814641296 |c hardcover : alk. paper |9 978-981-4641-29-6 | ||
| 020 | |a 9789814641302 |c pbk. : alk. paper |9 978-981-4641-30-2 | ||
| 035 | |a (OCoLC)927167449 | ||
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| 100 | 1 | |a Oliveira, Suely |e Verfasser |0 (DE-588)1076753485 |4 aut | |
| 245 | 1 | 0 | |a Building proofs |b a practical guide |c Suely Oliveira ; David Stewart |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2015 | |
| 300 | |a X, 164 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 0 | 7 | |a Beweistheorie |0 (DE-588)4145177-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Beweistheorie |0 (DE-588)4145177-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Stewart, David |d 1961- |e Sonstige |0 (DE-588)138178011 |4 oth | |
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| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028211800&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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Datensatz im Suchindex
| _version_ | 1804175010984624128 |
|---|---|
| adam_text | This book introduces students to the art and craft of
writing proofs, beginning with the basics of writing
proofs and logic, and continuing on with more in-
depth issues and examples of creating proofs in
different parts of mathematics, as well as introducing
proofs-of-correctness for algorithms. The creation of
proofs is covered for theorems in both discrete and
continuous mathematics, and in difficulty ranging
from elementary to beginning graduate level.
Just beyond the standard introductory courses on
calculus, theorems and proofs become central to
mathematics. Students often find this emphasis
difficult and new. This book is a guide to understanding
and creating proofs. It explains the standard moves
in mathematical proofs: direct computation, expanding
definitions, proof by contradiction, proof by induction,
as well as choosing notation and strategies.
Contents
Preface v
1. Getting started 1
1.1 A first example................................................ 1
1.2 The starting line: definitions and axioms...................... 4
1.2.1 Definitions............................................ 5
1.2.2 Axioms................................................. 6
1.3 Matching and dummy variables................................... 7
1.3.1 Matching up expressions................................ 7
1.3.2 Moving to the goal..................................... 9
1.3.3 Dummy variables ...................................... 11
1.4 Proof by contradiction........................................ 13
1.5 “If and only if ”............................................. 17
1.6 Drawing pictures.............................................. 18
1.6.1 Pythagoras’theorem — picture version.................. 18
1.6.2 Eulerian paths and the bridges of Koenigsberg......... 20
1.7 Notation...................................................... 22
1.7.1 Choosing names........................................ 22
1.7.2 Introducing new notation ............................. 23
1.7.3 Sub-scripts, super-scripts, and decorations........... 24
1.8 More examples of proofs* ..................................... 24
1.8.1 Composition of continuous functions................... 24
1.8.2 Sums of squares....................................... 26
1.9 Exercises .................................................... 27
vii
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Building proofs: A practical guide
Logic and other formalities
2.1 Propositional calculus.............................
2.1.1 Propositional logic and truth tables.......
2.1.2 Precedence.................................
2.1.3 Truth tables ..............................
2.1.4 Tautologies................................
2.1.5 Rules of inference.........................
2.1.6 implication via assumption*..................
2.1.7 Inference steps............................
2.1.8 The algebra of propositions................
2.1.9 Boolean algebra............................
2.2 Expressions, predicates, and quantifiers ..........
2.3 Rules of inference.................................
2.3.1 Rules of inference for propositional calculus
2.3.2 Rules of inference with quantifiers .......
2.4 Axioms of equality and inequality....................
2.5 Dealing with sets....................................
2.5.1 Set operations.............................
2.5.2 Special sets...............................
2.5.3 Functions.................................
2.6 Proof by induction ...............................
2.6.1 Some sums.................................
2.6.2 Fibonacci numbers..........................
2.6.3 Egyptian fractions........................
2.7 Proofs and algorithms.............................
2.7.1 Correctness by design* ...................
2.8 Exercises ........................................
Discrete and continuous
3.1 Inequalities......................................
3.1.1 Some classic inequalities ................
3.1.2 Convex functions .........................
3.2 Some proofs in number theory......................
3.2.1 Division algorithm and gcd................
3.2.2 Prime numbers.............................
3.2.3 “There are infinitely many primes” ....
3.3 Calculate the same thing in two different ways . . .
3.3.1 Sums of degrees in graphs.................
Contents ix
3.3.2 The integral exp(-z2) dx............................. 77
3 4 Abstraction and algebra...................................... 78
3.5 Swapping sums, swapping integrals ............................. 82
3.5.1 The exponential function .............................. 82
3.5.2 Taylor’s theorem with remainder........................ 84
3.6 Emphasizing the important...................................... 86
3.6.1 Equivalence relations and well-defined definitions ... 86
3.7 Graphs and networks............................................ 90
3.8 Real numbers and convergence................................... 96
3.8.1 The Least Upper Bound property......................... 96
3.8.2 Convergence of infinite sums........................... 98
3.8.3 Different kinds of convergence.........................100
3.8.4 “Give an e of room... ” 102
3.9 Approximating or building “bad” things with “nice” things . . . 103
3.10 Exercises .....................................................107
4. More advanced proof-making 113
4.1 Counterexamples and proofs.....................................113
4.1.1 Minimizing functions of one variable...................113
4.1.2 Interchanging limits...................................115
4.2 Dealing with the infinite .....................................116
4.2.1 Rearrangements of conditionally convergent series ... 116
4.2.2 Shift operators........................................118
4.3 Bootstrapping..................................................119
4.3.1 For square matrices, det(AB) — det(A) det(£?) . . . . 120
4.4 Impredicative definitions......................................121
4.5 Diagonal proofs................................................126
4.5.1 Russell paradox........................................127
4.5.2 Godel’s incompleteness theorem ........................128
4.6 Using duality..................................................129
4.6.1 Duality in linear algebra..............................130
4.7 Optimizing ....................................................132
4.7.1 Holder inequalities....................................133
4.7.2 Huffman coding....................................... 136
4.8 Generating functions...........................................141
4.8.1 Recurrences and generating functions...................141
4.8.2 Catalan numbers........................................143
4.9 Exercises .....................................................144
x Building proofs: A practical guide
5. Building theories 149
5.1 Choosing definitions ............................................149
5.2 What am I modeling?..............................................150
5.2.1 Generalizing known concepts.............................151
53 Converting one kind of mathematics into another..................152
5.4 What is an interesting question? ................................153
5.5 Exercises .......................................................155
Bibliography 159
Index 161
|
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| building | Verbundindex |
| bvnumber | BV042781777 |
| classification_rvk | SK 130 |
| classification_tum | MAT 036f |
| ctrlnum | (OCoLC)927167449 (DE-599)GBV822985608 |
| discipline | Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-10T07:09:30Z |
| institution | BVB |
| isbn | 9789814641296 9789814641302 |
| language | English |
| lccn | 2015012343 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028211800 |
| oclc_num | 927167449 |
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| physical | X, 164 S. Ill., graph. Darst. |
| publishDate | 2015 |
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| publisher | World Scientific |
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| spelling | Oliveira, Suely Verfasser (DE-588)1076753485 aut Building proofs a practical guide Suely Oliveira ; David Stewart New Jersey [u.a.] World Scientific 2015 X, 164 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Beweistheorie (DE-588)4145177-6 gnd rswk-swf Beweistheorie (DE-588)4145177-6 s DE-604 Stewart, David 1961- Sonstige (DE-588)138178011 oth Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028211800&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028211800&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Oliveira, Suely Building proofs a practical guide Beweistheorie (DE-588)4145177-6 gnd |
| subject_GND | (DE-588)4145177-6 |
| title | Building proofs a practical guide |
| title_auth | Building proofs a practical guide |
| title_exact_search | Building proofs a practical guide |
| title_full | Building proofs a practical guide Suely Oliveira ; David Stewart |
| title_fullStr | Building proofs a practical guide Suely Oliveira ; David Stewart |
| title_full_unstemmed | Building proofs a practical guide Suely Oliveira ; David Stewart |
| title_short | Building proofs |
| title_sort | building proofs a practical guide |
| title_sub | a practical guide |
| topic | Beweistheorie (DE-588)4145177-6 gnd |
| topic_facet | Beweistheorie |
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